Suppose you have a prime ideal $\mathfrak{P}$ of $\overline{\mathbb{Z}}$, the ring of algebraic integers in $\overline{\mathbb{Q}}$, that is lying over a rational prime $p\in\mathbb{Z}$, i.e. $\mathfrak{P}\cap\mathbb{Z}=p\mathbb{Z}$. Then the inertia subgroup of $\mathfrak{P}$, denoted by $I_{\mathfrak{P}}$ is defined as the kernel of the group homomorphism: $$ D_{\mathfrak{P}}\rightarrow\mathrm{Gal}(\overline{\mathbb{F}}_p\mid\mathbb{F}_p)\cong\overline{Z}/\mathfrak{P},\quad (\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}) $$ defined by $\sigma\mapsto\big(\alpha+\mathfrak{P}\mapsto\sigma(\alpha)+\mathfrak{P}\big)$, where $D_{\mathfrak{P}}=\{\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}\mid\mathbb{Q}):\sigma(\mathfrak{P})=\mathfrak{P}\}$ is the decompmosition group of $\mathfrak{P}$. In other words $I_{\mathfrak{P}}=\{\sigma\in D_{\mathfrak{P}}:\nu_{\mathfrak{P}}(\sigma(\alpha)-\alpha)>0, \forall\alpha\in\overline{\mathbb{Q}}\}$ where $\nu_{\mathfrak{P}}$ is the discrete non-arquemidean valuation associated to $\mathfrak{P}$ that extends the usual $p-$adic valuation on $\mathbb{Q}$ to $\overline{\mathbb{Q}}$.
Further suppose that $K$ is a finite extension of $\mathbb{Q}$ and that $\mathfrak{p}=\mathfrak{P}\cap\mathcal{O}_K$, where $\mathcal{O}_K$ is the ring of integers of $K$. In this case the inertia group $I_\mathfrak{p}$ is defined as the kernel of the group homomorphism $D_\mathfrak{p}\rightarrow\mathrm{Gal}(\mathcal{O}_K/\mathfrak{p}\mid\mathbb{F}_p)$, where $D_\mathfrak{P}=\{\sigma\in\mathrm{Gal}(K\mid \mathbb{Q}):\sigma(\mathfrak{p})=\mathfrak{p}\}$.
My question is the following:
If $\tau\in I_{\mathfrak{P}}$, why does there exist a positive integer $M$ such that $\tau^M\in I_{\mathfrak{p}}$?
I tried using the description of $I_{\mathfrak{P}}$ with $\nu_{\mathfrak{P}}$ because $\nu_{\mathfrak{P}}$ restricts to a discrete non-arquemidean valuation $\nu_{\mathfrak{p}}$ on $K$ which in turn is an extension of the $p-$adic valuation on $\mathbb{Q}$ and thus has an explicit formula depending on $\nu_p$ and the norm $N_{K_{\mathfrak{p}}/\mathbb{Q}_p}$, where $K_\mathfrak{p}$ is the completion of $K$ with respect to to $\nu_\mathfrak{p}$ . I think $M$ has to do with the ramification index of the extension $K_\mathfrak{p}/\mathbb{Q}_p$ , but I am not so sure. The question seems to be simple, so my problem might be in the definitions I am using (which is why I wrote them out).
I found the argument "$\tau\in I_\mathfrak{P}$ implies $\tau^M\in I_{\mathfrak{p}}$'' in a proof of a theorem that relates the semistablitity of an elliptic curve $E/\mathbb{Q}$ at a prime $\ell$ to the semistability at $\ell$ of another elliptic curve $E'/\mathbb{Q}$. Specifically proposition 7.1 of $\S7$ of Silverberg's chapter "Explicit Families of Elliptic Curves with Prescribed mod $N$ representations" in the excellent book "Modular Forms and Fermat's Last Theorem" edited by Cornell, Silverman and Stevens.